Anshel–Anshel–Goldfeld key exchange

Anshel–Anshel–Goldfeld protocol, also known as a commutator key exchange, is a key-exchange protocol using nonabelian groups. It was invented by Drs. Michael Anshel, Iris Anshel, and Dorian Goldfeld. Unlike other group-based protocols, it does not employ any commuting or commutative subgroups of a given platform group and can use any nonabelian group with efficiently computable normal forms. It is often discussed specifically in application of braid groups, which notably are infinite (and the group elements can take variable quantities of space to represent). The computed shared secret is an element of the group, so in practice this scheme must be accompanied with a sufficiently secure compressive hash function to normalize the group element to a usable bitstring.

Description

edit

Let   be a fixed nonabelian group called a platform group.

Alice's public/private information:

  • Alice's public key is a tuple of elements   in  .
  • Alice's private key is a sequence of elements from   and their inverses:  , where   and  . Based on that sequence she computes the product  .

Bob's public/private information:

  • Bob's public key is a tuple of elements   in  .
  • Bob's private key is a sequence of elements from   and their inverses:  , where   and  . Based on that sequence he computes the product  .

Transitions:

  • Alice sends a tuple   to Bob.
  • Bob sends a tuple   to Alice.

Shared key:

The key shared by Alice and Bob is the group element   called the commutator of   and  .

  • Alice computes   as a product  .
  • Bob computes   as a product  .

Security

edit

From the standpoint of an attacker trying to attack the protocol, they usually learn the public keys   and  , and the conjugated public keys   and  . A direct attack then consists of trying to find a suitable   that is generated by the elements of  , and that produces the appropriate conjugations   when applied. (An 'indirect' attack would consist of trying to find   directly, which would require some additional special structure of the group.) For this reason the public keys   and   must be chosen to generate a large subgroup of   — ideally, they form a full set of generators, so that   cannot be constrained just by knowing that is generated from  .

Solving for a suitable   given the conjugation relations is called the conjugation problem, and substantial research has been done on attacks to the conjugacy problem on braid groups, although no full efficient solution has achieved.

See also

edit

References

edit
  • Anshel, I.; Anshel, M.; Goldfeld, D. (1999). "An algebraic method for public-key cryptography" (PDF). Math. Res. Lett. 6 (3): 287–291. CiteSeerX 10.1.1.25.8355. doi:10.4310/MRL.1999.v6.n3.a3.

Further reading

edit
edit